Hi there. I have this exercise, which says:

Demonstrate that:

has a polynomial solution for some λ values.

Indicate the orthogonality relation between polynomials, the fundamental interval, and the weight function.

So I thought I should solve this using Frobenius method. I have one singular point at x=0, which is regular. I assumed a solution of the form:

And then replacing in the diff. eq. I get:

Therefore r=0.

Then replacing r=0, and changing the index for the first summation, with m=n-1, n=m+1:

And now calling m=n

So I have the recurrence relation:

Trying some terms:

I'm not sure what this gives, I tried this:

This is wrong, because the factorial in the numerator is only defined for positive values of (n-1-λ), and if n=1 I get -λ!, which wouldn't work for a_1, unless λ=0, which gives the trivial solution. But I think it works for n>1.

So I tried in a different fashion:

And now I called:

I think this is wrong too, because for example, n=1 gives which doesn't fit.

Then λ-n can't be a negative integer, and the polynomials would be given by:

Anyway, I took the diff. eq. into it's self adjoint form:

I actually think that I didn't have to get this explicit solution. To demonstrate what the problem asks I think I should take the equation to the self adjoint form.

Multiplying by

I get:

This is the self adjoint form for my differential equation. Then the weight function is given by:

I don't know how to get the fundamental interval.

Would it be ok just to work it in this last form, with out finding an explicit solution?

How can I get the fundamental interval?